。
She insists that she won't sing in front of anyone.
Literal
She [topic-は] anyone [genitive-の] front [at-で][even-も] does-not-sing [quotative-と] is-asserting.
誰の前でも is the universal-quantifier construction with でも attached to a question word: 'in front of anyone / no matter whose presence.' で marks the location, も adds the inclusive 'even / any.' 断言する is a strong verb meaning 'to assert categorically / declare without qualification' — much more emphatic than just 言う. The ~ている here is the resultative state of the assertion: she made the declaration, and that declaration stands.