Be sure the radio is not plugged in
when you do the following test.
I can assure you that if you take a flashlight battery (1-1/2 volt) and connect a small wire to one end of the battery and to the wired side of the speaker and the other side of the battery to the chassis you should get a clicking sound. If so then the chassis is acting as the return path from the audio transformer. There are usually four wires on an audio transformer, 2 on the primary and 2 on the secondary.
The secondary is always hooked to the speaker (usually 2 wires, one going to each connector on the speaker.) The company that made the radio was probably trying to save some money by not using extra wire. Just think in these terms, if you make 5000 radios and each radio took one foot of wire just for the speaker, you could cut your cost in half by using a existing connection, so only 1/2 foot of wire is used instead, the other missing wire is actually the chassis because its a conductor.
The squealing from the speaker is probably feed back from the phonograph. That says the speaker is hooked up correctly. This is caused by motor vibration and in turn being feed through the needle and amplified and again going through the same route over again and again. Most phonographs were isolated by springs from the main frame of the radio. When phonographs were shipped they were designed to be tied down with a screw on each corner, if this was not released then the sound from the speaker would travel through the wood frame back to the phono and the same thing would get amplified and then you would get a howl or squeal.
If the sound coming from the speaker is more like a low frequency hum that is quite loud then you have capacitors that are shorted or opened up in the radio. THEY MUST BE REPLACE, before you turn on the set again or you will damage the radio beyond repair. If you get sound but it is scratchy then you may have a speaker that needs to be repaired or replaced.
-----------------------
The wires on the back of the radio may be for an external antenna or may be broken off the tuning capacitor.
If you think you are going to repair this without an electronics background you are advised to learn the fundamentals and hazards of dabbling in things electrical, it could cost you your life.
Do you think you can work on a car without knowing or having basic automotive knowledge? Radios are the same way.
But if you are determined to try you need basic tools of the trade, soldering iron, solder, electrical test equipment, Hand tools and the ability to ask questions and be patient for answers and be willing to learn and get your self a basic book on radio repairs and trouble shooting. there are several on the web and a lot of helpful pages.
I will see if I can find the schematic for reference. It is in SAMS set 36 folder 17 which isn't available to me.
_________________
I'm a Science Thinker, Radio Tinkerer, and all around good guy. Just ask Me!