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Stonehedge-
I was curious and looked it up, and well, seems like you are correct, but also it's all a bit of a mess and the MGB is part of a trio of companies of which one is the infrastructure (owned entirely by government), one is the management (owned half by the government, half by BVZ Holding), and one is MGB itself (presumably the actual trains etc.? Not sure) which is mostly owned by BVZ Holding.
So... yeah, we got some complicated company structures there too.
SwissChap
According to their data, no - the company is fully held by the 'Bund' or the 'Schweizerische Eidgenossenschaft' (Swiss Confederacy), which is the government at federal level. By the way, there are other companies also providing rail services such as the BLS, owned over half by the Kanton Bern and over 20% by the confederacy.
(Oh, also, the SBB itself in turn owns a whole host of other companies either fully or partially, such as SBB Cargo, a catering company, and some regional railways.)