One speaker only works in Mono mode

Your (5.1) receiver supports just 6 speakers. Left, Center, Right, Left Surround, Right Surround and Subwoofer.
http://redirect.onkyousa.com/redirect.cfm?type=own_manuals&file=TX-SR304_En_0124.pdf

Some info from the manual:
The Dolby Digital and DTS listening modes can only be selected if your Blu-ray Disc/DVD player is connected to
the AV receiver with a digital audio connection (coaxial,
optical, or HDMI).
• The listening modes you can select depends on the format of the input signal. To check the format, see "Displaying Source Information"
• While a pair of headphones is connected, you can select the following listening modes: Mono, Direct, and Stereo.
• While Speakers A and B are on, you can select only the
Direct, Stereo, Mono, or T-D (Theater-Dimensional) listening mode.

Plus this Bit of info on the speakers A and B buttons.
While Speakers B is on, channels are reduced to 2.1 in the
main room.

If you can access the speaker configuration it should provide test tones to each speaker. If you can't the speaker A and B are not set right.
If you have pressed the stereo or mono button no sound will be heard except from the front two or the sub.

It may be the surround mode you are in. Some modes do not use the center channel. Try changing to a pro logic mode, these will almost always use the center

Could be the IC that controls the various sound/surround channel modes. Had that happen on my Pioneer receiver.
Charged me close to 200 bucks to fix!

If you put DVD with Dolby Digital soundtruck it will show on your TX-SR304 Dolby Digital mode, and if you have DVD with DTS mode it will show DTS

when the stereo or mono listening mode is selected only front speaker and subwoofer produce sound--------make sure that speakers are configured correctly.
and you select surround sound in listening mode.

I suspect that the speaker levels need to be set. You connected everything right - many people would not have connected the sats via the bass section (which has their high pass crossover and overload protection). But I am betting the HSU is turned up a bit on the high side.

Try this. Set all speakers to small, set all levels to 0. Play the receiver in stereo all mode. If the bass is strong, you need to turn the HSU down closer to a natural sound. Then use the speaker setup tone and adjust as needed.

Onkyo offers access to manuals at their web site. HERE is a link to the receiver line.

Personally, I'd connect the speakers to main left/right. The subwoofer to subwoofer out. The Go VIdeo unit to VCR/Video 1, the cable box to Video 2 and the video output to the Sony TV composite input. No need to connect the TV audio out to the receiver since the audio feed comes from the cable box.

It's going into a protect mode. The amp is probably shorted out. Disconnect the spkrs and turn it on. If it still shuts off, it's the amp, if not, it's the spkrs.