In a word, no. First of all, the “new program” isn’t really a new program at all – it’s an extension of a program that Obama first signed into law in 2012, known as the “Pay As You Earn” program (or PAYE). Originally, the 2012 version of PAYE allowed only *new* student-loan borrowers to pay 10% of their discretionary income for 20 years before the remaining balance, if any, is forgiven.
Recently, Obama made news yet again for expanding the PAYE program to other Federal Direct Loan borrowers who are not new borrowers. It is unclear at this point exactly which borrowers of older loans will be covered by this extension of the PAYE program, just that the borrowers need to be borrowers of Federal Direct Loans.
So unless you have borrowed a Federal Direct student loan, Obama’s new program will not help you, especially not with a private student loan. The only potential new legislation out there affecting private student loans are the various proposals in Congress that have virtually no chance of passing.