Is there a definition for it?
What should an Islamic country's government do so that we call it Islamic?
In what way is a Islamic government different from any other government?
If Islam is defined as being truly submitted to Allah, then an Islamic country is one that has some characteristics:
its governing rules are the rules declared by Allah
its ruler is chosen by Allah
Otherwise, according to Shi'ite (as much as I have understood) the government will be called Taaqut and it is forbidden for Muslims to obey such a government unless in a state of Taghiyyeh (reservation?).
Indeed, Allah is the Lord (المالک الملک), the only King (الملک، ان الحکم الا لله), and no one can rule on His behalf on His Earth except with His specific permission. So that even a prophet that is not chosen by Allah as a ruler if rules he will be an example of Taaqut, like that Allah says:
Their Prophet said to them: "Allah hath appointed Talut as king over you." They said: "How can he exercise authority over us when we are better fitted than he to exercise authority, and he is not even gifted, with wealth in abundance?" He said: "Allah hath Chosen him above you, and hath gifted him abundantly with knowledge and bodily prowess: Allah Granteth His authority to whom He pleaseth. Allah careth for all, and He knoweth all things."
If this prophet PBUH was introducing himself as a ruler for the Israelite then he was an example of Taaqut, but so perfect that Allah is He knows whom to choose for the prophecy, someone that will never disobey Him! However, Talut was also ruling Israelite with the rules of Allah and not rules of his wish. Then David and Solomon PBUT were chosen as rulers and they were also prophets at the same time. In Shi'ite's believe an example for those who were true believer Muslims but at the same time Taaqut is Umar-ibn-Abdul-Aziz, for he did a lot for Muslims (and even Shia Muslims) while he was ruling the Islamic societies but at the same time he was Taaqut because he was not eligible in the view of Allah for this ruling position. In a Hadith from Imam Baqir-al-Ulum AS the narrator tells that Imam pointed toward Umar-ibn-Abdul-Aziz in Mosque and murmured that he is the one after whose death people on the Earth will cry for him passed away but angels in the sky will not, as he has sat on a seat not belonging to him. This was to illustrate the importance of the must for the ruler (be him a true believer or even a prophet or whatsoever) to be chosen by Allah for the country governed by him or his government to be Islamic.
If this condition is met it will be trivial that such a ruler will try his best to also exploit the rules of Allah in his country, so that the other condition would be trivial as already mentioned.
According to Quran:
Muslims are forbidden to undergo non-Islamic rules:
… And never will Allah grant to the unbelievers a way (to triumphs) over the believers. (An-Nesaa':141)
Muslims are even ordered not to resort to Taaqut for the judgement:
Hast thou not turned Thy vision to those who declare that they believe in the revelations that have come to thee and to those before thee? Their (real) wish is to resort together for judgment (in their disputes) to the Evil One, though they were ordered to reject him. But Satan's wish is to lead them astray far away (from the right). (An-Nesaa':60)
Muslims are not even allowed to take such Taaqut governments as their supporter and protector:
Or think ye that ye shall be abandoned, as though Allah has not flagged those among you who strive with might and main, and take none for friends and protectors except Allah, His Messenger, and the (community of) Believers? But Allah is well-acquainted with (all) that ye do. (At-Tubah:16)
And all these altogether imply that, if possible, Muslims should have their own government and the ruler should also be coined the title "ruler" by Allah, then this government will be an Islamic government and the country governed by such a government will be an Islamic country.
Also you may like these verses:
Hast thou not turned Thy vision to those who were given a portion of the Book? they believe in sorcery and Evil, and say to the Unbelievers that they are better guided in the (right) way Than the believers! (An-Nesaa':51)
Never will the Jews or the Christians be satisfied with thee unless thou follow their form of religion. Say: "The Guidance of Allah,-that is the (only) Guidance." Wert thou to follow their desires after the knowledge which hath reached thee, then wouldst thou find neither Protector nor helper against Allah. (Al-Baghareh:120)
Islamic country and Islamic government are completely different.
Islamic country is a country that majority of its people are Muslim.
Shia Islam View
Islamic government is a government using Islamic laws are base of the rules and plans of the government and the best form of it is that a top Jurist be the top governor of the country like government of Prophet Muhammad (SAWW) at Medina or Government of Imam Ali (a.s.) at Kufa.
The best source in this regard is the famous book Islamic Government: Governance of the Jurist by Imam Khomeini (ra) which currently is the base of Iran Islamic government after Iran Islamic Revolution.