I know that repairs in sheet vinly can be done but I don't know about vinyl tile squares. I'd be inclined to think they need to replaced just like sheet vinyl.
If you're repairing sheet vinyl, visit How To Repair Sheet Vinyl. It pretty much sums it up.
Other than that, maybe others have more to offer. My best to ya and hope this helps.
PS: After this afternoon, I won't be back until Monday, 9/18 but 'others' will be around ...
© 2017 Renovate Your World LLC