Can my sister’s boyfriend kick her out of an apartment that they have shared for 6 years?

While your sister should consult with an attorney to be certain of her rights and recourse, the unfortunate fact is, if she is not on the lease (if the apartment is rented) or is not on the title (if the apartment is owned), then she may not have any rights to the apartment or to reside there. If it is his apartment, he can evict her unless she is actually in some way renting from him, in which case she will at least enjoy the protections any tenant would. If she has been contributing to the rent, mortgage, utilities, etc., it might be the case that she would at least be considered a "tenant at will" or "month to month" tenant, which would give her at least one month's notice before eviction commences.

If she's contributed to purchasing an apartment--or accumulating or buying any assets--she might be able to establish some equitable right to them and at least receive some compensation for her contribution.

Also, if she sacrificed or gave up her career in order to do things that materially furthered his career or business, it's also possible--not certain, but possible--she may be able to establish some claim based on promissory estoppel or unjust enrichment.

There are therefore potential claims that are worth investigating, which is why your sister should consult with a lawyer. However, none are particularly strong claims--if someone is not on a lease or deed, does not have an agreement or contract, is not married, and has simply relied on another person without legal ties or obligations, she is in a very vunerable position.