My Question is About Chin Implants from 35 Years Ago. I Believe It Was 1978. I'm Having Discomfort?

For part of the fact to have discomfort,
there needs to be a source of inflammation, infection or other condition to
cause the symptoms of discomfort. It would be very unusual for a chin implant
to have such issues such as infection so many years after the implant was
inserted unless there was some disturbance in the anatomy such as trauma.

Chin
implants are mostly manufactured out of substances that are well tolerated by
the body long-term. Chin implants can be made out of a variety of materials,
the most common being silicone, and less commonly alternative materials such as
hydroxyapatite (Medpor). The silicone used in chin implants is solid silicone
and this does not deteriorate with time and can be removed years later. It is
possible to complete imaging (CT or MRI) of your face to evaluate for localised
inflammation surrounding the implant and local anatomy. There may be other
causes of discomfort for neighboring anatomy near the chin implant included
dental, salivary glands and lymaphtic notes that may need an evaluation to
exclude another cause of discomfort.

Chin implants are made of solid silicone, unlike breast implants, which have been made with liquid silicone.  The solid silicone would not break down or travel throughout your system, so I doubt it would be the cause of issues in your neck.  

As the other respondents indicated, it is highly unlikely that your chin implant is causing a problem, as long as it was made of solid silicone rubber.  However, there were other materials used in the past made of other substances than can break down.  However, these usually would not cause gland swelling.  It is much more likely that your gland swelling is being caused by chronic inflammation of the submandibular glands, and not the implant.  An Otolaryngologist (Ear, Nose & Thoat doctor) can easily assess you for this.

While your old chin implant may be some form of silicone, it would be highly unlikely to be leaking and causing any automimmune reactions. If you really want to see whether your chin implant is causing any local reactions, a simple panorex x-ray looking for any osteolysis (bone erosion) around the implant would be a good screening tool.