I heard that squeezing the toothpaste tube might be considered wringing. Does this have any basis?
Some say yes, some say no.
The issue is not only wringing, but also whether the paste turning into froth is considered nolad, something new that came to be on Shabbat. For a comprehensive overview of the issues involved and which poskim have ruled what, see Yabia Omer Vol. IV O"H #30, where Rav Ovadia rules that it is permissible to brush your teeth using regular toothpaste on Shabbat.
For a summary of that responsum, here is the Yalkut Yosef 326:14:
Some Poskim ruled that even someone who brushes teeth with toothpaste every day may not do so on Shabbat for various reasons. One reason given for this is that when one brushes, the toothpaste becomes frothy with suds, and therefore this is an act of molid. Another reason given is that vigorous brushing might cause the gums to bleed.
Other Poskim ruled that it is permitted as long as one places the toothpaste directly into one's teeth and not on the toothbrush. Yet other Poskim ruled that even more leniently, permitting one to brush one's teeth on Shabbat just as one does on the weekdays and with regular toothpaste. They explained that it is not necessary to use special, liquid toothpaste, since brushing teeth is not an act of the melakha of memare'ah, nor is it an act of molid. Furthermore, this is not forbidden as an example of using medical treatments on Shabbat.
In practice, anyone who would feel uncomfortable if he does not brush his teeth on Shabbat because of bad breath or some other reason may brush them. This applies, however, only to someone who regularly brushes his teeth so that he will not necessarily cause his gums to bleed on Shabbat.