Daenerys also had her own prophecy. Consider the dragon dream she had before her wedding:
She closed her eyes and whimpered. As if in answer, there was a hideous ripping sound and the crackling of some great fire. When she looked again Viserys was gone, great columns of flame rose all around, and in the midst of them was the dragon. It turned its great head slowly. When its molten eyes found hers, she woke, shaking and covered with a fine sweat.
and again on the Dothraki Sea:
There was only her and the dragon. Its scales were black as night, wet and slick with blood. Her blood, Dany sensed. Its eyes were pools of molten magma, and when it opened its mouth, the flame came roaring out in a hot jet. She could hear it singing to her. She opened her arms to the fire, embraced it, let it swallow her whole, let it cleanse her cleanse her and temper her and scour her clean. She could feel her flesh sear and blacken and slough away, could feel her blood boil and turn to steam, and yet there was no pain. She felt strong and new and fierce.
This prophecy was fulfilled in Chapter 72 of AGOT when her dragons were born. At this point she knows that at least some magic is real and she may be more inclined to believe in other prophecies.
The Mirri Maz Duur quote itself is not exactly positive - it is more of a challenge to Daenerys. That Drogo will never return until these impossible events happen. And she heard this during a very memorable series of events, between Drogo's death, the sacrafice of Mirri Maz Duur, the birth of her dragons, etc.
The essay Under the Bleeding Star has an excellent analysis of the role of prophecies across the series and succinctly describes Dany's attitude towards this one:
When Mirri Maz Duur spits these words in Dany's face, she doesn't take them as a prophecy, but thinks it just means "never."
Yet it is not surprising that Daenerys would think back to these events, especially considering that Mirri Maz Duur was the first challenge that Daenerys had to face on her own without the help of a "leader" such as Viserys or Drogo. So perhaps you could say that she accepted the challenge, rather than blindly trusting the prophecy of an enemy.
As to the Undying, Daenerys drank shade of the evening beforehand, which put her in a more impressionable state of mind:
One draught will serve only to unstop your ears and dissolve the caul from off your eyes, so that you may hear and see the truths that will be laid before you.
And she knows some of the visions that she sees in the house are true:
- Daenerys's childhood home with the red door in Braavos.
- Viserys's gruesome death.
- Daenerys's silver trots through grass to a darkling stream under a sea of stars.
Perhaps she sees a part of herself in these visions, or thinks they are from her as much as the Undying? Maybe she trusts in their magic? Hard to say - I agree it is suspicious that she trusts their words given that they try to kill her right after. Then again, if they were going to kill her anyway, why even show her the visions? Maybe she thinks the Warlocks uncovered prophecies they did not like, and chose to kill her to prevent them from taking place.
Unfortunately we never get a definitive answer - but given her experiences I think it is plausible that she would at least consider these prophecies and think back on them rather than disregarding them outright.
For what it's worth, Under the Bleeding Star weighs in on this prophecy as well:
She's constantly wondering whether the prophecy has already been fulfilled or not. Her decisions are impacted greatly by her fears of its wording - who might be the next betrayer? She believes she can't avoid the prediction, but perhaps she can take away some of its impact.
When you consider that the "three treasons" are as much a curse as a prophecy, it makes sense that she would try to minimize the impact of that curse.
Finally, as an out-of-universe explanation, perhaps GRRM is trying to tell us something. We know that he foreshadowed many important events during the House of the Undying chapter that have already come to pass. And there are many more yet to happen, if GRRM's "prophecies" from the chapter are to be trusted.