The short answer is no, it wouldn't. The main reason for that is that squirting in and of itself does not create an STI risk. It may help to read up on what squirting is and is not: Squirt: On Ejaculation
Too, if a condom is being used for all instances of sex, that in and of itself is doing a lot to lower STI risks, especially risks coming from fluids like vaginal secretions or ejaculate. Really, there are only a few STIs that can transmit even when a condom is being used, and you can read about those here: HPV & Herpes: Why Safer Sex Isn't Always Safe Enough
Since it sounds like STIs are something you're worried about, are you and your partner up to date on your STI testing?