That’s an interesting question. Your two examples have slightly different structures, so different considerations apply to verb agreement.
1. "Neither the Company nor any of its Subsidiaries have/has ..."
Two important points:
(a) A "neither ... nor" phrase comprises a marker (neither) + coordinator ("nor"), so this example displays a coordination of two noun phrases joined by the coordinator "nor".
(b) "Any" occurs freely with both singular and plural agreement, cf. "Do you know if any of these cars is/are for sale". "Any of its subsidiaries" is a partitive construction in which "any" is called a 'fused determiner-head'. It means "Any subsidiary (one or more) from the set of its subsidiaries", so "any" can have a singular or plural interpretation. Note, by the way that "any" is head of the noun phrase, not "subsidiaries". The effect of this is that "any of its subsidiaries" can be conceptualised as 'one subsidiary' or 'one or more' subsidiaries’, so either singular "has" or plural "have" is acceptable.
2. "Neither the Company nor any Subsidiaries have / ?has ..."
Things are different here. "Any subsidiaries" is unarguably plural, so we have a nor-coordination with a singular noun phrase and a plural one. Nor-coordinations of coordinates with unlike numbers are problematic, and the singular verb "has" is of questionable acceptability, especially when the nearest coordinate to the verb is plural, as it is here. I'd stick with plural "have" if I were you.