Yes, in theory. But;
No, this is very unlikely to happen during the life of the sun. If it did happen it would be Bad.
Many black holes are much more massive than the sun, even though they are much smaller. If a black hole were to enter the sun, some the matter that forms the sun would form into a accretion disc around the black hole and eventually fall into the black hole. The rest of the sun would be utterly disrupted by the massive release of energy as matter falls towards the black hole.
The solar system would be wrecked. The Earth, even if it survived intact, would not be able to support life.
Fortunately black holes are rare, and space is BIG. The chances of a black hole coming anywhere near the sun in the next billion years or so is very very small. There are more pressing things to worry about.
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