Jampots - a lawyer here to clarify.
As a general principle, yes you can definately breach a breached agreement - it depends on what clause was breached and the exact terms of the agreement.
What you have to consider is whether the agreement has been terminated by the first breach (or by subsequent cancellation). If the first breach terminated the agreement, then there is no agreement to breach (and any subsequent "breaches" are immaterial, because there is no longer an agreement to breach). In your case, where your employer has simply not paid (as I understand it this is the breach on his part that you are referring to) the agreement would probably not terminate, but would need to be enforced. This means that you must take care not to breach any of the terms that relate to you. If both parties to an agreement make material breaches, then a court may well decline to enforce the agreement.
Not sure what breach you have (or intend to) commit, though. If you are referring to your employer's allegation that you breached confidentiality of the agreement - well, that is for him to prove (and you say you didn't, so he would not be able to prove it). Sounds like he is just trying to scare you.
Difficulty that you have, of course, is proving that an agreement exists (and on what terms). However, the copy (with your signature on it) together with your employers email saying that he has countersigned and returned the agreement to you, and the documents from the tribunal should be enough to prove that an agreement was reached and on what terms (and can be enforced).
Hope this helps.