I think you're talking about SIGSALY, which was a system used for trans-oceanic radio communication only at the highest levels (FDR, Churchill, Eisenhower, et al). It was impractical for much else as it filled a room and weighed 50 tons. Since cell phones now use similiar technologies, you can guess that they've since learned to make these things a bit smaller.

Flairs: please understand that this is a history of technology subject (sorry, but I've seen similar questions deleted as not pertaining to history).

You used the term "static" in your description. It didn't transmit static (radio noise), but it sounded that way before it was decoded. Frankly I don't know where "squelch" would be used in it's description, as in radio that refers to silencing the audio when no one is talking.

As for the the history other than the history of the technology, that's not my area of expertise. I've already mentioned it was used only at the highest levels. Otherwise I'll mention that the system was developed by Bell Labs, and was first used in 1943. If you want more information on the non-technological aspects of the history, hopefully someone else can provide that. I'll give a brief overview of the technology. The sources at the end give more detailed information if you want (and some cool pictures of the equipment), and cite sources which are more detailed yet.

SIGSALY used a number of technologies which were developed specifically for it, or improvements on technologies that had been developed just a few years before. It started with a vocoder (voice encoder) which takes audio and converts it into a compressed digital form. "Compressed" simply means that it takes fewer bits to represent the audio. Vocoder technology is still used for cell phones and the like, albeit much improved. With SIGSALY the audio was divided into ten frequency bands, and every 20 milliseconds the level of the of each band was measured and quantized to 6 bits (which can represent 64 different levels). Thus it used one of the first digitized audio systems. Next came the encryption part, where the digitized audio was combined with a random sequence of digital data, thus rendering it unreadable to anybody who didn't know the random sequence. Both sides of the communication had the same random sequences recorded on phonograph records, so they could decrypt each others signals. Lastly, the information was turned into a form that could be transmitted by radio using FSK (frequency shift keying), which was yet another new technology. Basically FSK translates different levels into different frequencies. In SIGSALY the last two steps were combined into one, which is different from what you'd do today, but is conceptually the same.

Sources:

Audio Engineering Society, "SIGSALY"

NSA Historical Publications, "Sigsaly - The Start of the Digital Revolution"