To a degree it was, but everything I've seen has portrayed US involvement as relatively minor compared to the other participating powers.

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First of all, let's get some context. During the regime of Haile Selaissie, Ethiopia was a close ally of the US. The Ethiopians received arms and sent officers abroad for training, while the US was given access to the radio/SIGINT installation at Kagnew Station in Eritrea. Ethiopia had been dealing with unrest in Eritrea during this time, but by the end of Selaissie's regime, Ethiopian forces largely had the situation under control. However, Selaissie's Ethiopia had several stark parallels to the situation in Iran, with the regime attempting to rapidly modernize and westernize while keeping a rapidly aging absolutist monarch at the helm. The clash of Western liberalism and an ancient monarchy went about as well as one would expect - a failed coup was launched in 1960 in an attempt to force a transition to a more accountable government, only for Selaissie to double down when it was crushed. As Selaissie only became less competent as he aged, the situation boiled over again in the mid-70's with a more Marxist flavor to it. Selaissie and much of the royal family were imprisoned and either died in prison or were executed. The Derg - a committee of army officers - regime that took over initially did what it could to retain the US presence and support, but as the regime took an increasingly Marxist flavor and the situation deteriorated into purges and a Red Terror, the US began to wean off its support, ultimately withdrawing from the country entirely and shutting down Kagnew Station in 1977.

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Meanwhile, the Soviets had their eyes on the region as well, hoping to get both an ally in the strategically important shipping chokepoint and a military port on the Indian Ocean. They ended up getting Somalia under the regime of Siad Barre to fall into the Soviet sphere, trading arms and aid for the creation of a Soviet naval base at Berbera. Soviet-backed Somalia largely leveraged its aid to push irredentist claims in neighboring countries, often through sponsoring terrorist groups like the West Somali Liberation Front in Ethiopia. When the Derg took over in Ethiopia, the Soviets saw this as an opportunity to create a united Soviet bloc in the Horn of Africa region. Castro's Cuba became involved, and attempts were made to foster cooperation between Barre's Somalia and the Derg's Ethiopia. Castro even went as far as to propose a united Horn of Africa, bringing Somalia and Ethiopia together into a single federation.

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Unfortunately for the Soviets, Barre had other plans. By 1977, the situation in Ethiopia seemed near collapse. The rebellion in Eritrea sparked back up, tying down the Ethiopian army. Purges had hit the officer corps hard, depriving Ethiopia of much of its American-provided experience. And the US had finally cut off the supply of arms, leaving the Ethiopian army short on equipment. Perhaps the best indication of the disarray in Ethiopia in 1977 was the confusion in the opening days of the war with Somalia - as Somali tanks and aircraft streamed into Ogaden and wiped out isolated Ethiopian garrisons, the Ethiopians took several weeks to realize that they were being invaded and that this wasn't just a particularly aggressive offensive on the part of the WSLF. I'll leave out the details of the war for now, but the war caused a huge backlash towards Somalia. The Soviets and Cubans demanded Barre withdraw his troops and cut off military aid. The Somali army stagnated short of its goals, and Barre, desperate for supplies, made the decision to eject the Soviets from their recently completed naval base at Berbera. Barre hoped to quickly flip sides to the US bloc, but found that the Carter administration was very reluctant to provide anything but token support for a nation still locked in a war of aggression that it had launched without warning. Barre did look elsewhere for help, getting some aid from other anti-Soviet states including Communist China and Egypt, but military aid was not coming nearly as fast as Barre needed.

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The Soviets, meanwhile, had responded rapidly to the loss of Berbera. They launched the largest airlift effort the Soviet airforce had done since the Second World War, bringing in hundreds of tanks and other vehicles, 50,000 Cubans and hundreds of advisors, tens of thousands of tons of materiel, and even allegedly diverting a satellite launch to provide a better handle on the situation on the ground. After some preparation, the Soviet and Cuban-backed Ethiopians pushed the Somalis from their country, nearly annihilating the Somali army as it did. From that point on, Ethiopia remained firmly in the Soviet bloc, while Somalia wouldn't see the kind of support Ethiopia was receiving nor the kind it enjoyed from the Soviets from any of its new patrons. Somalia would nominally be a US ally, but it wouldn't recover to become a major player in the region. Ultimately, a lot of the replacement equipment Somalia would receive following the war was Chinese rather than American - the post-1978 Somali Air Force was provided Chinese J-7 (MiG-21 clones) fighters, replacing the Soviet-supplied MiG-21s it had during the war.

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The massive Soviet and Cuban presence in such a strategically important region was a concern to the US, however. National Security Advisor Brzezinski claimed that “SALT lies buried in the sands of the Ogaden," as the rapid Soviet involvement was the first of several aggressive moves that caused the US to withdraw from the SALT II talks, and the presence of a huge number of Cuban troops and an aggressively Marxist regime in such a strategically important region was a major concern to the US and neighboring Arab states. Foreign involvement was most active in the Eritrean theater, with the Eritrean-Sudanese border region becoming a refuge for rebels owing to Sudanese support. However, as far as I've read, the US never had the kind of active support for rebels that we saw in other proxy wars like Angola, nor did it have agents in the region it could count on to move aggressively for similar interests like it did with South Africa and Zaire. The US would only support the Eritrean Liberation Front (ELF) until the end of the Carter administration, and as they were supplanted by the more communist Eritrean People's Liberation Front (EPLF), open support for the rebel groups in Eritrea and Ethiopia only really materialized once the writing was on the wall and the war had effectively already been won. Support from US-aligned Arab states was more overt, however. Sudan, Egypt, and Libya were actively supportive of the Eritrean rebels, and, as in Somalia, Communist China was happy to undermine Soviet interests in the region by providing support as well.