It seems likely that costs associated with the material prevents economical production of red glass. Rather than a matter of possibility, it may be a matter of viability. Without knowing more about how the glass is produced or the specific materials used for their bottles, it is difficult to say the exact reason. The best answer I can think of is that the only glass I am aware of which selectively transmits red light requires gold nanoparticles suspended in the silica glass network. Such glass is referred to as Cranberry Glass or Gold Ruby Glass (Wikipedia).
It isn't clear how much gold is required, though the abstract of this paper, which uses an alternate production process, suggests about 0.2% by weight. The price of gold is about 40,000 USD/kg per the current spot price from apmex.com as of 08 March 2016 at ~2:45 PM CST. Thus a glass bottle weighing 1 kg would have an added cost of about 80 USD from the gold alone. Expanding the process to mass-produce such bottles would require storing and shipping significant quantities of gold, which may add logistical and overhead costs for security. There would also be equipment requirements for producing colloidal aqua regia in the proper concentration and drying, handling, and mixing the nanoparticles into the liquid glass.
As a side note, flint glass (Wikipedia) contains zirconium and/or titanium. I can't think of any reason that either element would interfere with the cohesion of the gold particles, but I also can't strictly rule it out. If such a thing did happen, then Flint glass specifically might be impossible to make red by the use of gold nanoparticles.