One would presume that Morgoth would have the power, but is there any hint Sauron could have broken it as well?
As far as I know Sauron never personally tests himself against the Girdle, so the best we can do is look at the types of beings who might be able to break it. According to The Silmarillion (emphasis mine):
Melian put forth her power and fenced all that dominion round about with an unseen wail of shadow and bewilderment: the Girdle of Melian, that none thereafter could pass against her will or the will of King Thingol, unless one should come with a power greater than that of Melian the Maia. And this inner land, which was long named Eglador, was after called Doriath, the guarded kingdom, Land of the Girdle.
The Silmarillion III Quenta Silmarillion Chapter 10: "Of the Sindar"
It's been said before on this site that the idea of relative strength is ill-defined in Tolkien's Legendarium, so it's difficult to say definitively "yes" or "no". However, we know that Sauron was the greatest of the servants of Morgoth:
Among those of [Morgoth's] servants that have names the greatest was that spirit whom the Eldar called Sauron, or Gorthaur the Cruel. In his beginning he was of the Maiar of Aulë, and he remained mighty in the lore of that people.
The Silmarillion II Valaquenta "Of the Enemies"
We also know that the Girdle is remarkably resilient; it possibly turns back Ungoliant, who is hopped up on the sap of the Two Trees to the point where she poses a credible threat to (an admittedly-exhausted) Morgoth himself:
[Ungoliant] quailed, and turned to flight, belching black vapours to cover her; and fleeing from the north she went down into Beleriand, and dwelt beneath Ered Gorgoroth, in that dark valley that was after called Nan Dungortheb, the Valley of Dreadful Death, because of the horror that she bred there.
The Silmarillion III Quenta Silmarillion Chapter 9: "Of the Flight of the Noldor"
Nan Dungortheb, we're told later, lies just on the edge of the Girdle:
There it was put into [Beren's] heart that he would go down into the Hidden Kingdom, where no mortal foot had trodden. Terrible was his southward journey. Sheer were the precipices of Ered Gorgoroth, and beneath their feet were shadows that were laid before the rising of the Moon. Beyond lay the wilderness of Dungortheb, where the sorcery of Sauron and the power of Melian came together, and horror and madness walked. There spiders of the fell race of Ungoliant abode, spinning their unseen webs in which all living things were snared; and monsters wandered there that were born in the long dark before the Sun, hunting silently with many eyes.
The Silmarillion III Quenta Silmarillion Chapter 19: "Of Beren and Lúthien"
If anyone was going to break the Girdle, it would probably be Sauron; even in the First Age he's among the more powerful of the Maiar. Then again, Melian is also an unusually powerful Maia, one of the greatest (if not the greatest) of those who lived in the land of Lórien before the Coming of the Elves:
Melian was a Maia, of the race of the Valar. She dwelt in the gardens of Lórien, and among all his people there were none more beautiful than Melian, nor more wise, nor more skilled in songs of enchantment.
The Silmarillion III Quenta Silmarillion Chapter 4: "Of Thingol and Melian"
Unfortunately, in the absence of a direct confrontation, it's difficult to argue definitively.