Yes, the Squadron Supreme was explicitly intended to be a version of the Justice League of America. However, the JLA encountered a version of the Avengers in their own book at the same time.
As best I recall, the writers of the books at the time planned the thing (without explicit permission from either company).
At Marvel, the Squadron Sinister had appeared in 1969 (AVENGERS 69-70, October-November 1969); the characters were villains based on prominent DC characters (Hyperion [Superman]; Nighthawk [Batman]; the Whizzer [the Flash]; and Dr. Spectrum [Green Lantern]).
Nighthawk appeared again shortly thereafter in DAREDEVIL 62 (March 1970), but on the whole it seemed like the characters were intended as a one-off thing.
However, in February 1971, Marvel published AVENGERS 85; and, that same month, DC published JUSTICE LEAGUE OF AMERICA 87.
The Avengers travelled to another dimension and met a heroic version of the Squadron Sinister, named the Squadron Supreme, with the four characters mentioned above, joined by others: Lady Lark [Black Canary], Hawkeye/Golden Archer [Green Arrow], Tom Thumb [Atom], and American Eagle [Hawkman]. The story continued in the next issue.
Meanwhile, the Justice League travels to another world, Angor, and meets the Heroes of Angor: Jack B. Quick [Quicksilver], Silver Sorceress [Scarlet Witch], Blue Jay [Yellowjacket], and Wandjina [Thor].
The Squadron Sinister popped up occasionally as villains in Marvel Comics. Nighthawk turned against the team (he didn't like the idea of destroying the planet, for some odd reason), and became a mainstay in the original Defenders team until his apparent death (DEFENDERS 13-106). However, years later it turned out he was just in a coma, and he got better. The other member of real note is the Whizzer (not to be confused with Marvel's WWII-era speedster). Evidently, he eventually tired of urination jokes, and changed his name to Speed Demon, becoming a member of Spider-Man's rogues' gallery, and of the Sinister Syndicate; he was used to good effect a few years ago in SUPERIOR FOES OF SPIDER-MAN.
The Squadron Supreme also popped up on occasion, sometimes as allies, sometimes as opponents. They are perhaps best remembered for their mid 1980s SQUADRON SUPREME 12-issue miniseries, which is probably remembered less for its story than for the writer's ashes being added to the ink after his death. Their later appearances continued to echo the membership of the JLA, with characters like Amphibion [Aquaman], Zarda (aka Power Princess) [Wonder Woman], Skymax the Skrullian Skymaster [Martian Manhunter], Arcanna [Zatanna], and Nuke [Firestorm].
The Heroes of Angor have also appeared again, but not nearly so frequently. They were brought back to the Justice League books by JM DeMatteis and Keith Giffen during their run on the series (affectionately known as the "Bwa-ha-ha" run). I think most of them died during that run (it wasn't all jokes); it's possible that Blue Jay has appeared a couple of times since then.
There have been two newer versions of the Squadron Supreme. J. Michael Straczynski created a new version of the team for Marvel around 2003, under the title SUPREME POWER or SQUADRON SUPREME. Most of these were published under Marvel's MAX imprint, which generally publishes more "adult" material (i.e., work with gory violence and/or sex/nudity). Between the two titles and various related mini-series, a total of around 60 issues were published (not all by JMS, to be clear).
The other newer version came out of the Jonathan Hickman AVENGERS run, and the 2015 SECRET WARS story. This was a collection of people resembling the JMS version of the team, but from different worlds that were destroyed (at least one from an Earth destroyed by Namor the Sub-Mariner in Hickman's NEW AVENGERS) during the destruction of the Marvel Multiverse.