Can I power both legs of my breaker panel with 120 volts from a Honda eu3000is generator?
Yes you can power both legs of the panel from a single 120 volt source. Any 240 volt appliance won't work properly, and as you noted multi-wire branch circuits would be at risk of over-current on their neutral. Everything else should be fine.
One thing to check is the neutral-ground bond. Honda EU-series do not have a built-in neutral-ground bond. Make sure your system is appropriately bonded when switched to generator inlet mode.
Edit: More info about neutral-ground bond
Part of the reason we bring ground wires along with the line and neutral wires, at least in the USA wiring custom, is for safety. If certain kinds of electrical faults occur they could result in current flowing from one of the energized lines to earth-ground. Earth (dirt) isn't a great conductor, though, so the current won't be high enough to cause a circuit breaker to trip. Tripping a breaker is a good thing because it de-energizes the fault and because it calls attention to the problem. It turns out that if we &quot;bond&quot; (connect) the earth-ground wiring to the neutral then that fault current will be high enough to cause a breaker to trip.
There should be only one neutral-ground bond in a system. In a single-family home context that bond should be found in the first service panel -- usually the one where the electric meter is installed. It can take several forms: a screw that electrically connects a neutral bus bar to the housing of the panel, or a jumper wire between neutral bar and ground bar, or simply neutral wires and ground wires all landing on the same bus bar.
By definition all transfer switches switch the line conductors(s). Some also switch the neutral (and some switch the ground too? I'm not certain.). If a system has a 1- or 2-wire transfer switch, ie it switches only the line conductors, the generator connection should be arranged so that it does not introduce a second bond. With a 3- or 4-wire transfer switch, though, one needs to figure out whether the existing bond becomes disconnected by operation of the transfer switch.
In summary: it comes down to tracing wires. When the house is configured for normal grid power confirm electrical continuity (0-ohm resistance) between ground and neutral wires. Figure out where that connection/bond exists. Then configure the house for generator power, trace wires, and figure out whether that same ground-neutral bond is still in effect (and is the only bond), or whether there are 0 or 2+ bonds in the generator configuration.