Perfect conditional

'She shouldn't have told him.'

The context of this sentence is, "It was a secret, she shouldn't have told him".

To analyse the form of target sentence - 'she' - is the subject, 'him' is the object, 'have told' - is perfect infinitive. What is 'shouldn't'?

My second question, is this analysis correct?
Third question, why is it 'shouldn't' and not 'wouldn't'. I know 'shouldn't is correct, but why.

Thank you for your time.