Just to add to Gina Myers' very helpful response, Francis Crossle and his son Philip were professional Irish genealogists. Their papers are now available on FMP. Some documents are for their reference - copies of lists of deeds, bills, wills etc - and some are working papers in files for research they did for different clients. I get the impression that they were extremely meticulous, better than people like Burke, but you have to check whether what you are looking at is their finished research or just something they copied from someone else for information. As Gina rightly notes, this pedigree is just copied from someone else, although even then the Crossles have added a note with their own doubts.
The pedigree itself mentions a few more facts about Thomas Phillips than Gina has noted. It has something (which I can't read) dated 1662 in a place which is probably Belturbet. This is a town in county Cavan where the people you ask about are supposed to have been based. Then there is a note (not clear whether copied from the original or added by the Crossles) that Thomas Phillips was MP for Clonmines co Wexford (in quite a different part of Ireland) in 1692. The records certainly confirm that a Thomas Phillips was MP there on that date. Then it shows that he died in c 1700 having written a will dated 12 July 1699. I don't see a will for Thomas Phillips proved in 1700, but others were proved on dates around then including one for Thomas Phillips of Belturbet in 1692, or rather I think that this Thomas died intestate and an administration bond was granted to his brother William. Interestingly, William does not feature in the pedigree that Gina dug out, but does feature on Wikitree which states him to have been one of the six children of Sir Thomas mentioned in his will, citing this paper in the extremely reputable Irish Historical Studies. However, the citation is false, as the paper itself says that the six children mentioned in his will were Dudley, Elizabeth, Dorothy, Chichester, Thomas and Anne - which is what the pedigree has. (It is a Canterbury Court will, avalable on Ancestry.)
Looking at these points and the Crossles' comment, I wonder whether two or possibly thee different people have been merged into one in the pedigree. Sir Thomas certainly had a son Thomas not yet of age when he died in 1636, but whether this Thomas was the person who married Jane Hannah Richardson, was the MP or was the one in Belturbet I don't know.
As to Thomas junior having been a Quaker, I don't see any evidence of it at all, and the person who was an MP certainly couldn't have been one.
