Could cossacks carve out independant states outside the Russian Empire. This is not to be confused with carving out states from the Russian Empire.
Which time frame and which "Cossacks"?
Russian Empire existed between 1721 and 1917 and by that time the window of opportunity was mostly gone.
Term "Cossacks" is so broad that it does not make too much sense. Initially, there were Ukrainian and Don Cossacks. The Ukrainian ones had a subdivision, "Zaporozie cossacks" (permanently existing group of the bandits making their living mostly by fishing and raiding). The Ukrainian Cossacks got their own initially independent state, "Hetmanate", in the mid-XVII (no "Russian Empire" around for quite a while) but it soon was split into lesser entities controlled by the PLC, Tsardom of Moscow and Crimean Khanate (situation was quite fluent for years). The main problem was an absence of the good natural borders. Well, and general disorder within that state(s). However, the Hetmanate (vassal of Moscow/Russian Empire) existed as an ...er.... "independent state" all the way to the reign of Catherine II. Its last Hetman (junior brother of a boyfriend of Empress Elizabeth) had its capital (but preferred to live in St-Petersburg), court, etc. When Catherine decided that she had enough of that silly notion, he protested and was banished from the court. In a very short time he understood all errors of his ways, agreed to all Catherine's demands, was forgiven, patted on the head, allowed to kiss the empresses hand and lived happily ever after. Zaporozhian Sich was abolished in 1775.
The Cossacks of Don had been from the very beginning too dependent upon the supplies from Moscow (firearms, gunpowder, etc.) to try playing the independence. During the Time of Troubles they were, of course, doing their share of looting and there were some rebellions in the XVII and early XVIII but that's it.
The Cossack of Yaik (or Ural Cossacks) had been a driving force of the Pugachev Rebellion but, short of their leader claiming to be Peter III, there was not too much of a "statehood" involved.
The Cossacks of Siberia (there were numerous groups) hardly had a chance or wish because there was too few of them even put together.
The Central Asia did not provide realistic openings: local population was bellicose enough, not enough Cossacks had ever been in that region and it was probably too late for such an idea, anyway.
What about non independant cossack hosts outside the Russian empire?
During the reign of Catherine II quite a few of the Ukrainian/Zaporozie cossacks fled to the Ottoman empire. There was even a comic opera, "Zaporozets beyond the Danube" dedicated to this event. The Ottomans formed
Danubian Sich in Bessarabia which existed until 1812.