My Panasonic TH-60CX700A does not pair with my Bose sound link headphones

Likely you wont be able. It is like trying to pair two headphones together. Both units are designed to receive sound but not to send out

notify manufacturer or seller

The design of these headphones includes a microphone on the outside and on the inside of each earcup. The outside microphone detects outside noise. The inside microphone monitors what you are hearing and when it detects that the outside noise is present inside, the active noise-canceling circuits remove that noise. This happens fast enough that you perceive a significant reduction in the outside noise getting to your ears.

The airspace inside the earcup is a tuned chamber defined by the shape of the foam in the ear cushion. This is important because with the inside microphone in close proximity, but at a 90-degree angle to the speaker, the circuit must be designed to eliminate the normal feedback that occurs.

If the foam in the ear cushion has weakened/deteriorated to the point that it no longer has it's original shape, (in particular the lower half of the ear cushion), the size and shape of the air chamber inside of the earcup is directing more of the speaker sound toward the microphone than the design can compensate for causing audio feedback to occur. You are hearing that feedback.

On my QC-3 headphones, I can cause the feedback squeal in both earcups, with the left earcup being more sensitive to the shape change than the right.

It is probably time to replace the ear cushions. If so, they are available from Bose for $35/pair.

QuietComfort 3 Acoustic Noise Cancelling Headphones Bose Product Support

They are also available aftermarket at Amazon.com for $16.99/pair
Replacement Earpad ear pad Cushions For Bose QuietComfort 3 QC3 Headphones...

or $5.69/pair.
Bluecell Pair of Replacement Earpad Ear Pad Cushion for Bose Quietcomfort...

The ear cushions are held on the earcup with six snap fasteners, three on each side. To remove the ear cushion, simply pull it away from the earcup carefully so that you do not break the plastic snap tabs.

It should not be necessary to pry the ear cushion off with a sharp object, but if you do, be careful. There is also a thin gasket seal between the earcup and the ear cushion, take care not to damage it.

If you need to replace the ear cushions with new ones, they are available from Bose for $35/pair.

QuietComfort 3 Acoustic Noise Cancelling Headphones Bose Product Support

They are also available aftermarket at Amazon.com for $16.99/pair
Replacement Earpad ear pad Cushions For Bose QuietComfort 3 QC3 Headphones...

or $5.69/pair.
Bluecell Pair of Replacement Earpad Ear Pad Cushion for Bose Quietcomfort...

Does this mean you have picture? If yes then did you try connecting a pair of headphones into the headphones jack to see if sound? This should help us check if its a problem with the speakers or the audio/video mainboard.

If your headphones are in warranty (under a year old) Bose will repair or more likely replace them with a new pair. I am now on my forth pair or Bose in-ear headphones and experienced the same problem with an ear piece stop working. I have to say Bose customer service has been fantastic and has replaced each pair. Every new pair is then covered by a new full years warranty. Unfortunately my fourth pair has just gone wrong with the same connector problem. Bose again have said they will replace as they are in warranty. I have been told if they are out of warranty they will offer a new pair at a discount price of £50 instead of £80. However i think that if my new pair go wrong out of warranty i will have no option but to look at alternatives.A Pair of headphones that cost £80 should last more than a year!!!

They will give you a new pair for $100, even if it has been longer than a year...

http://riajournal.com/ria/index.php/2009/04/25/bose-gets-5-stars-in-my-book-for-their-qc3-headphones/

Yeah, based on all of your problems, i would safely say that you have a blown speaker. Call up Bose at 1-800-999-2673. If the headphones are under warranty (1 year) they will replace them for free. If now, they will charge you $100 for a replacement pair, which is quite a deal considering that the headphones cost $300.