The Lack of A coast line hampers this immediately.
Even if this Switzerland did have a coastline and resembled OTL Switzerland, the big issue would be disunity. Up until the 1848s, Switzerland resembled a miniature HRE. The Cantons had signifigant autonomy, as they were basically sovereign nations who were bonding together to avoid The Hapsburgs' influence (among other things).
Now in this Wanked-out Coast-possessing Switzerland, an individual canton could possibly set it's sight for getting a colony, but now it has to tread lightly, for the Venetians, Spaniards, Brits, etc. will seize the colony to simply push a weaker country out of 'their zone'.
Though Brandenburg and a few other HRE states (IIRC) had colonial companies, they left very little impact and made little to no gains. All of the basically gave up the ghost by the 1680s...except for the Dutch.