Would breaking a rule in the handbook constitute breach of contract?
Generally speaking, yes. Of course, most employment contracts are "at will" so a breach of contract often isn't necessary to terminate employment.
Does my employer's updating the handbook constitute a unilateral
change of contract?
It depends upon what is changed in the update to the handbook.
Typically an employment contract would have a term that says something to the effect of "employee shall devote his full time effort to perform the duties he is directed to perform by employer in a satisfactory manner."
If the employee handbook says, "part of every employee's duties includes cleaning up his work area at the close of business each day, locking his file cabinets and shutting down his computer", this would typically simply be a definition of the duties of the employee which the employer reserved the right to change in the original employment contract, and not a unilateral change of contract.
On the other hand, if the employee handbook says, "vacation days may not be taken during December" when the employee's contract simply said that "you have ten vacation days per year", that might constitute a unilateral change of contract which might not be enforceable for an employee with a fixed term of employment who was not an employee at will, without additional consideration.
Locale is UK but (I assume) contract law is pretty universal so
answers specific to any region are welcome.
I have answered based upon general contract and employment law, but the UK frequently sets mandatory standards for different kinds of contracts and modifies common law rules related to contracts (much more so than the U.S.), so it wouldn't be very surprising if this were modified by a statute of which I am not aware.
We have UK lawyers who contribute to Law.SE and they can chime in if there are particular statutes in the UK that apply to this question.