Any relationship between thyroid disorders or the medication to treat thyroid disorders and spider veins/broken capillaries?

There isn't a known correlation between thyroid disorders & spider veins. Venous insufficiency can cause spider veins, due to backed up blood or they can simply be genetic in origin. I would suggest scheduling a vein screening appointment to have a treatment plan individualized and to see if there is underlying insufficiency. 

There is a definite interplay between thyroid disease and the metabolism of the body. I am not familiar with direct links, but there may be indirect ones. Read more about risk factors on our site. Best, Dr. KaramanoukianLos Angeles vein expert

There should be no relationship between thyroid disorders and the development of spider veins.

Thyroid disease is not known to cause spider veins, reticular veins, varicose veins or venous insufficiency and a PubMed search done on 1-7-2017 does not show any results. However, this does not mean that inflammatory processes in the body, like thyroiditis, does not accelerate inflammatory venous disease. I would wear compression stockings and I recommend microfiber graduated 20-30 mm Hg knee high or thigh high stockings from VenaSmart. Hratch L Karamanoukian MD FACS certified vein specialist (ABVLM) 

There is no obvious relationship between thyroid disease and spider veins.  Spider veins develop due to reflux in adjacent superficial veins.