Is ''blurt out'' synonymous to ''jump the gun''?

They flow along the same lines in a vague manner. But one main difference between the two is that the former is inadvertent, in most cases, usually provoked by fear, anxiety, etc. The latter, if you were to look up its synonyms and usages, you would know that it is mostly used in situations that are hasty. You could say prematurely revealing something.

blurt out:

to utter with a sudden burst of strong feeling

jump the gun

to do something too soon, especially without thinking carefully about it:

Hence the two big differences:

1. one can only SAY things while blurting out, while jumping the gun can be any action
2. jumping the gun must be too soon, while blurting out can be at any time.

For instance, if someone was keeping a secret from another person for a bad reason, he could suddenly realize that this will cause a lot of problems and blurt the secret out, without jumping the gun -- indeed, while being vastly tardy in telling it.