SuttaCentral
To clarify my own position, I didn’t start this thread with the intention of assessing or critiquing Jain beliefs. I don’t know enough about them to do that. So I’m not taking any position on what Jains believe or practice. I merely wanted feedback on Bronkhort’s idea that there might have been Jain influences in the suttas, especially in regard to the passage I quoted.
Buddhism’s relationship with ascetic practice (the dhutangas) is an interesting topic, but wasn’t what I intended to discuss.
If you happen to be interested in Jain influence in mahāyāna, there is quite a bit in the tathāgatagarbha sūtras. The aṅgulimālīya sūtra is a good example of this. The tathāgatagarbha is a lot like the Jain view of ātman, and it promotes abstaining from flesh, although it has passages critiquing Jain practice as suttas do.
This article should also be of interest, it discusses parallels and similar passages in early texts. 
Bronkhorst is an interesting case of great erudition and, when it suits him, mixing texts from very different time periods to make his point. I’m afraid this is such a case where he’s referencing Jain texts which clearly don’t belong to an early stratum. Hence, it isn’t suitable to draw a connection to the Buddha’s time.
I should mention, for the volume I provided the link for above, that one probably won’t be of much interest as it deals with comparison of language and dialect mostly. The first volume is far more interesting as it deals with the contents of the suttas themselves.
Can you provide the free download link version? 
These types of threads that go into connections between Jainism and early Buddhism are always kind of difficult. This may be in part because there is so little surviving early textual evidence for Jainism. But a few things may be mixed together:
It’s also hard to discuss asceticism because it can mean anything and everything from simply eating once a day, to some pretty extreme and harmful practices that would be objectionable to most.
The excerpt from the Mahasaccaka Sutta may not be referring to practices from Jainism at all. While Bronkhorst does claim that the passage from the Mahasaccaka Sutta describes mainstream meditation, all his other examples in which the tongue being pressed against the palate are from texts belonging to the Brahmanical tradition. And Hindu yogis are still the ones that do this. I have not seen examples of this practice being advised in Jainism.
Yes, you have a point. It’s known as khecari-mudra
The MN 12 Mahasinada Sutta indicated Jain practices done by Bodhisatta before his enlightenment (although the Pali commentary said it is during one of his previous lives, not in his last live):
And this is what my self-mortification was like. I went naked, ignoring conventions. I licked my hands, and didn’t come or stop when asked. I didn’t consent to food brought to me, or food prepared specially for me, or an invitation for a meal. I didn’t receive anything from a pot or bowl; or from someone who keeps sheep, or who has a weapon or a shovel in their home; or where a couple is eating; or where there is a woman who is pregnant, breastfeeding, or who has a man in her home; or where food for distribution is advertised; or where there’s a dog waiting or flies buzzing. I accepted no fish or meat or liquor or wine, and drank no beer. I went to just one house for alms, taking just one mouthful, or two houses and two mouthfuls, up to seven houses and seven mouthfuls. I fed on one saucer a day, two saucers a day, up to seven saucers a day. I ate once a day, once every second day, up to once a week, and so on, even up to once a fortnight. I lived committed to the practice of eating food at set intervals.
I ate herbs, millet, wild rice, poor rice, water lettuce, rice bran, scum from boiling rice, sesame flour, grass, or cow dung. I survived on forest roots and fruits, or eating fallen fruit.
I wore robes of sunn hemp, mixed hemp, corpse-wrapping cloth, rags, lodh tree bark, antelope hide (whole or in strips), kusa grass, bark, wood-chips, human hair, horse-tail hair, or owls’ wings. I tore out hair and beard, committed to this practice. I constantly stood, refusing seats. I squatted, committed to the endeavor of squatting. I lay on a mat of thorns, making a mat of thorns my bed. I was committed to the practice of immersion in water three times a day, including the evening. And so I lived committed to practicing these various ways of mortifying and tormenting the body. Such was my practice of self-mortification.
MN 12 seems to show that the Buddha previously carried out every type of extreme ascetic practice. But he never describes himself as a Jain (Niganthaputta). The ascetic practices were associated with developing supernormal powers, but MN 12 is making the point that they didn’t develop the wisdom of liberation.
MN 12 describes Gautama’s practice as a brahmacarya with four angas, and lists those four. MN 56 and DN 2 describe the practices of Nigantha Nataputta (Mahavira) as having a fourfold restraint, but those seem to be unrelated and with little overlap.
Going back to MN 12, can most of the practices be traced back to the Jains specifically? Going naked is one obvious practice of some Jains, as is non-harming… But, for example, do we know that Jains were wearing owl wings or tree bark? What about antelope hide? Wasn’t wearing furs and sitting on furs associated with Brahmin ascetics and hermits?
Wearing tree bark, living in the forest, and eating wild fruits is a typical description of the sramanas (according to Megasthenes). But do we know that Jains specifically engaged in those practices, or that those practices were characteristic of the Jains?
This list refers quite clearly to ‘wanderers of other sects’ as a collective term. Niganthas were either naked or wore white garments.
Ajina, the black antelope hide was indeed the ceremonial dress used in the Vedic diksa consecration ritual which appears already in the Taittiriya-Samhita. The yajamana sponsor of the ritual wore it only for a short time though. In the Atharvaveda it is the brahmacarin who wears the antelope skin, probably for a longer time. Later it became synonymous with an ascetic attitude and was probably taken over by ascetic professionals - but not by Jains.