Nope, I don’t buy it.
This amendment, as I see it, says that we, as citizens cannot be required or forced by government to house military personnel in our private residences against our will.
It has nothing to do with anyone NOT paying his/her rent, military or not.
"No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation."
This is effectively taking private property for public use without just compensation. The Government can order no evictions, but must compensate the property owners if rent is not received from the tenant.