At that point they then face the most heavily armed populace on the face of the earth on their home turf, and there would be massive casualties on both sides, probably eventually leading to an insurgency of sorts by the american people.
That geography is important, you have to control those choke points to move tanks, artillery, troop convoys, supplies, and without those bridges, places where highways converge, and tunnels there's no way to support an army moving further into the states (all of these places tend to have military bases within 20-30 miles of them as well).
pic related, /k/ has threads on this occasionally that are gold but they dont delve into tactics too often>>84283990
nah its a fun thing to ponder on>>84284154
yeah the river is crazy strong where its not wide and where its wide it's fucking WIDE. those crossing points are pretty much the only crossing points, everywhere is too unstable for bridges from the mississippi river being a fucking titan
and in the event all of america has a common enemy, you can bet folks would form up into militias pretty quick, the biggest problem would be keeping the cities fed so those populations dont start going all escape from NY