The supplication at the time of fasting is most likely to be answered, because it is at the end of an act of worship, and in most cases, a human self is at its weakest point just before he breaks his fast. The weaker a person is physically, and the more tender-hearted, the greater is his feeling of repentance and humility towards Allaah, the Almighty, the All-Powerful.
The supplication which is reported from the Prophet sallAllaahu ‘alayhi wa sallam is:
Çááåã áß ÕãÊ æ Úáì ÑÒÞß ÃÝØÑÊ
(Allaahumma Laka Sumtu Wa ‘Ala Rizqika Aftartu.)
“Oh, Allaah! For Your sake I have fasted and with Your sustenance I have broken my fast.”
When he broke his fast, the Prophet sallAllaahu ‘alayhi wa sallam also said:
ÐåÈ ÇáÙãÃõ¡ æÇÈÊáÊ ÇáÑæÞ æ ËÈÊ ÇáÃÌÑ Åä ÔÇÁ Çááå
(Thahabath-Thana’u Wabtallatil-‘Urooqu Wa Thabatal-Ajru In Sha’ Allaah.)
“Thirst has gone, the arteries are moist, and the reward is sure, if Allaah wills.”
Although there may be some weakness in these two Ahadith, some of the Scholars have declared them to be Hasan; and in anycase, if you supplicated using these words or any others words that came to your heart when breaking your fast, then it is most likely to be answered.
As for answering the Mu’aththin when a person is breaking his fast, it is lawful, because of the saying of the Prophet sallAllaahu ‘alayhi wa sallam:
ÅÐÇ ÓãÚÊã ÇáãÄÐä ÝÞæáæÇ ãËá ãÇ íÞæá
“If you hear the Mu’aththin, then say what he says.” 
applys to all situations, except those which there is some evidence to show otherwise.
 Reported by Abu Dawud in the Book of Fasting.
 Reported by Abu Dawud in the Book of Fasting, in the Chapter: What to Say When Breaking the Fast (2357).
 Reported by al-Bukhari in the Book of Athaan, in the Chapter: What is Said When the Caller is Heard (611); and by Muslim in the Book of Prayer, in the Chapter: the Recommendation of Saying What the Mu’aththinSays (384).