NO - because they are weapons and weapons don't kill, people do.
YES - because they take the money and then become the weapon AND as long as they do not question the task they are performing, just actting they take on the aligment of their user. You, then get the line "I was just following orders.
The exact same could be said about assassins - they are weapons "wielded" by someone else and they, too, are "just following orders".
And that's assuming they are the Assassins Guild "Hire us to make your problems go away" type rather than the government-trained "sneak in and take out our nation's enemies while they're miles from the battlefield"* (whose motivation may well be patriotism) or religious "kill the infidel" types.
* Which to my mind is a far more important "defining characteristic" than anything else mentioned so far.
A soldier's theatre of operations is the battlefield - be (s)he sniper or mercenary or militia or Regular Force - the assassin's theatre of operations is not constrained in that manner.
A mercenary or soldier must take on the enemy in the battlefield - and if they want that battlefield to be your capital city, they have to get there in force.
An assassin - paid, government, religious zealot or whatever - is "free" to enter areas other than a war zone and "inhume" their selected targets - while they are asleep, eating dinner, speaking at a public venue, having a crap, walking the dog, shagging the missus - whatever.
It is that departure from the designated war zone (wherever that might expand to during the course of the war) that differentiates between the assassin and the soldier.
In the case of an assassin, there doesn't even have to be a declared "shooting war" in place.
An assassin may act on the battlefield but a soldier who acts outside the battlefield has just crossed the line from "combat" to "assassination".
When soldiers turn up in your Capital, they turn up in numbers and try to fight their way to your leader.
An assassin sneaks in and waits for the opportunity to kill.