From the rules : The rules

Join the Military. When passing through this square, Unemployed and
Employees may choose to stop on this square and join the military,
with the following results:

The rules clearly state
«When passing through this square… may choose to stop…»

Joining the military is therefore NOT compulsory for Employees and the Unemployed; you can just move on, i.e. «pass through».

Since this rule applies only to the specified players, one can logically deduce that the other players, Owner and Manager, do not have this choice. [Edit : The author has confirmed this. See below]

The copyright holder of the game must then decide which of the alternatives these latter players must take. I have contacted Randal Sean Harrison, Ph.D. at the University of Notre-Dame and asked him to give a ruling.

Edit :
The author, Dr. Randal Sean Harrison, has answered in a private email.

It’s totally optional. Certain player classes [Employee and Unemployed] “may choose” — is the key phrase.
Managers and Owners may not join the military—as the rules were
written.

I suppose you could change that rule, but why would you? What
benefit would it provide the owner? In Landlord, that mechanic is a
vehicle for rising socioeconomically. I don’t know many business
owners or MBA or capitalist billionaires who join the military, etc.
🙂

Dr. Harrison also gives further information concerning play involving the «Join the Military» square that is not germane to your question but I expect would be extremely useful; so here it is:

Certain player classes “may choose” — is the key phrase. The idea
being that, say, I’m an unemployed player, am 2 squares away from
Military square and I roll a 6. I move the two squares and stop on the
military square. I exchange my unemployed card for employee. I’m not
an enlisted man/woman. I move now with a D6 (as an “employee class)
-1, since, the military restricts movement in ways civilian life does not. > And until I pass the Payday square next, I roll an additional D6.
If that roll is a 1, I roll again. If I roll again, I’m killed in
combat.

When I play this with students, some choose to join, some don’t.
Without college or the military, though, it’s nearly impossible to
rise up through the socio-economic classes. There are change cards
which will allow you to rise up, but they are few and the odds of
striking them are slim. Another fun fact. Every class of four boards
with 4-5 students, 1 or two students who join the military die. It’s
1/36 chance after all.

All that said, the most important way this is NOT monopoly (sic) is that
you can and should make rules (analog for politics). You (the player) can
remake the world. So the ultimate arbiter of the Military square
question is the player in the game in the moment.

In my estimation, this is the ultimate game for Monopoly players who like house rules. The way I understand it, the author’s objective is to get you to tweak the game rules so that you become more aware of what policies (rules) are overpowering certain social classes and what is required to make a more level playing field for all.

Also, the game’s name is «The Landlord Game», and not «The Landlord’s Game», the latter of which is Elizabeth Magie’s 1904 game.

The Landlord’s Game 1904