Nevil Wylie's Britain, Switzerland and the Second World War (Oxford University Press, 2003) is a good source on this, and does not mention any attempts by either side to press Switzerland to take an active part in the war.
The Swiss would have been extremely un-receptive to any such suggestions, given that neutrality had been the guiding principle of their international relations for centuries. Joining the Allied side when they were surrounded by the Axis powers would have been obviously suicidal. Most of the Swiss population were quite unsympathetic towards Nazism, which regarded them as properly part of Germany, while they felt they were Swiss, which is different.
Hitler loathed the Swiss political system, which is a rather individualistic form of democracy. This is the antithesis of Nazism, which specifically rejects democracy in favour of a one-party state, and regards individuals as important only in that they contribute to the goals of the state and propagate their "blood." Hitler would not have been willing to ally with the Swiss, and they would not have accepted being subject to his authority. The Wehrmacht made plans to invade Switzerland, in Operation Tannenbaum, but never put them into action.
The Swiss would not have that much to contribute to either side. The Swiss military is carefully designed for defending Switzerland, and not threatening its neighbours, which would greatly limit its usefulness in a continental-scale war.
Both sides had more to gain from Switzerland remaining neutral, although both sides wanted a kind of neutrality biased towards their interests. Both sides used Switzerland for espionage, and for secret contacts with the other side. The Allies used Switzerland's Protecting Power services, since large numbers of their servicemen were prisoners in a Europe running low on food, and as a refuge for escaping prisoners. Both sides bought machine tools, electrical equipment, and precision mechanical and optical instruments from the Swiss, although that trade was greatly restricted for the Allies once France was fully occupied in late 1942. The Germans imported dairy products, electricity, and various other manufactured goods, and made use of Switzerland's rail system for supplies to Italy. They also made use of Switzerland's Protecting Power services. Both sides made use of Swiss financial services, especially buying and selling gold. Swiss co-operation with Germany was reduced considerably in the final year of the war, especially once Allied troops reached Swiss borders.
Switzerland didn't have any major natural resources that could attract an invader and weren't already being exploited efficiently. Contracting out military R&D to a neutral country wasn't an idea that would have been used at the time, and it would have been "un-neutral conduct," which the Swiss always tried hard to avoid any appearance of, while avoiding being invaded.
Dealing with those pressures was the primary task of Swiss politicians during the war, which they managed adequately, if not perfectly.