First of all, please accept my apologies for the delayed response.
You inquired regarding the historical basis of the Exodus.
I am not aware of any hard evidence that contradicts the Exodus narrative. Much of what passes for archaeological knowledge is interpretation and hypothetical reconstruction of events based upon scant physical evidence. And there are certainly models of the data put forth by scholars in the field that would militate against the Biblical version of what occurred. However, there are competing interpretations and perspectives that support the traditional view, and I believe that a truly objective investigator would be foolish to reject them in favor of more iconoclastic alternatives.
In my opinion, rejecting the historical tradition of a nation is only warranted when there is no recourse, i.e., when evidence clearly refutes the claims of that tradition. Otherwise, a tradition unanimously held by an entire people for centuries should enjoy the benefit of the doubt, even if this means favoring interpretations of the data that are consistent with, rather than contradictory to, its claims.
So, in a nutshell – yes, the Exodus happened as written, and the proof is that an entire nation maintained this historical memory and based its religious beliefs and practices upon it for centuries, coupled with the fact that the Exodus narrative – while, strictly speaking, not provable – can be reconciled with plausible interpretations of the archaeological record.
I hope you find this answer helpful.