If a borrower is eligible for IBR payments, but then chooses to make a payment in excess of their calculated IBR monthly payment, does that payment go towards the principal (or does it, in any way, negatively impact their IBR eligibility)?

A borrower may pay more than what is due under IBR, just the same as under other repayment plans.  There is no prepayment penalty and it doesn’t affect eligibility.  But the payment is first applied to late charges and collection fees (if any) then to outstanding interest and only after that to principal. Lots of borrowers will have unpaid interest under IBR (unlike other repayment plans, it allows borrowers to pay less than the interest accruing) so it could take a big payment to have any attributed to principal.