We commonly say that sets obey a Boolean algebra. I think that's correct as a stand-alone statement. If true, did Cantor come up with a Boolean algebra on his own, or did he use the work of Boole?
Grattan-Guinness has an informative book on the history of logic and mathematical foundations, The Search for Mathematical Roots, 1870-1940. There are dedicated chapters on Boole and Cantor, and Boole is mentioned in Cantor's chapter not once. The history of set theory by Ferreirós returns the same null result.
Boole's work was only publicized in Germany in the late 1870s, when Cantor was already well on his way with set theory. Schröder's exposition Operationskreis des Logikkalkuls came out in 1877. This said, Schröder was a prominent mathematician with interest in set theory and foundations, so Cantor might have been aware of it, but if so he left no indication to this effect. On the reception of Boole's work in the 19th century generally see Klement, New Logic and the Seeds of Analytic Philosophy.
Cantor had little use for set algebra and did not introduce it in his work. When one needs a fact about sets that a Boolean identity might provide it is more straightforward to deduce that fact by a direct element argument (compare deriving some Boolean identities from axioms, which can get quite intricate, to plain element arguments for them). So unless one is specifically focused on algebraic properties of set operations they would not be interested in Boolean algebra of sets. Cantor was not.
In addition to that, Boole framed his work as about "laws of thought" and translating that into Cantor's context would have required clarifying obscure relationships between properties (truth conditions), classes (intensions) and sets (extensions), which would have been a major distraction. It was not done until the early 20th century, see How did mathematicians notate the empty set before ∅? "In fact, by the 1890s Cantor considered set theory as something completely different from logic", writes Ferreirós (Dedekind was far more receptive). Cantor also had a special distaste for "laws of thought", heavily associated with Aristotle at the time, for those "laws" were traditionally used to rule out actual infinities, see Why did Cantor (and others) use c for the continuum?