Any battery drains over time even if it is placed in a vacuum where electrons can't travel, because all known battery constructs have internal leakage. Lithium-ion in general is actually a lot better than many other technologies in this respect. The problem arise with rechargeable cells which shouldn't be discharged below a certain voltage, or they will loose their ability to take charge. To avoid that the equipment, and sometimes the batteries themselves, have a built-in circuit-breaker. A small amount of energy will leak through this circuit, but it is minuscule in most modern battery designs, and certainly insignificant compared to the load of an operational or even "sleeping" variax.
A 1/4" jack used as a power-switch should be a stereo-jack, and wired so that neither the amp nor the magnetic pickups become part of the circuit. I.e. any leakage should happen within the internal circuitry. The variax supposedly has powersaving-features, unlike a plain guitar with active pickups so they don't compare at all. I've got other guitars with active circuitry that uses the jack as power-switch, but there's next to no drain on the battery with the jack plugged if the built-in circuit has powersaving features or other internal power-switching (hardware or software).