* Migrants demand passage to Germany, Western Europe
* Hungary says enforcing EU rules on passports, visas
* Ruling party figure: we want no European Caliphate (Adds ruling party comments)
By Krisztina Than
BUDAPEST, Sept 1 (Reuters) - Hundreds of angry migrants demonstrated outside Budapest's shuttered Eastern Railway Terminus on Tuesday, demanding that they be allowed to travel on to Germany, as a migration crisis put the European Union's rules under unprecedented strain.
Hungarian authorities closed the train station altogether, then reopened it but barred entry to the migrants. About 100 police wearing helmets and wielding batons guarded the station. Dozens of migrants who were inside were forced out.
Around 1,000 people waved tickets, clapping, booing and hissing, and shouting "Germany! Germany!" outside the station. Later they sat down, staring down at a police blockade at the entrance of the station.
One man held up a sign that said, in German: "Please let us go!"
The arrival of hundreds of thousands of migrants has confounded the European Union, which no border controls for travel between 26 countries of its Schengen area but requires those seeking asylum to remain in the country where they first arrive until their applications are processed.
The vast majority of those arriving first reach the continent's southern and eastern edges and are determined to travel across Europe and seek asylum in more generous countries further north and west.
Hungary is on a major overland transit route from the Middle East and Africa for refugees who flee violence and economic migrants escaping poverty, with more than 140,000 people crossing its southern border with Serbia this year alone.
Hungarian and Austrian authorities allowed trainloads of undocumented migrants to reach Germany on Monday. Czech police said they had detained 214 mostly Syrian migrants headed for Germany on overnight trains from Vienna and Budapest.
The crisis has prompted the government in Budapest to reinforce its border with a razor wire fence and deploy thousands of extra police to try to funnel the flow of migrants to legal channels rather than allowing them through unchecked.
Faced with the enormous pressure of thousands upon thousands of migrants arriving in Budapest, Hungary let them board westbound trains on Monday before unexpectedly shuttering the train station again on Tuesday morning.
Government spokesman Zoltan Kovacs, when asked why the railway terminus was closed, said Hungary was trying to enforce EU law, which requires anyone who wishes to travel within Europe to hold a valid passport and a Schengen visa.
Hassan, a 47-year-old Syrian, said he and two friends had each bought tickets to Germany for a total of 370 euros.
"They took 125 euros for each ticket to Munich or Berlin, then they stopped and forced us from station," he said. "(They) said station is closed. They said no trains, this station is closed."
Marah, a 20 year-old girl from Aleppo, Syria, who travelled with her family, said they had bought 6 tickets for a RailJet train that was scheduled to leave for Vienna at 9 a.m. on Tuesday.
"They should find a solution," she told Reuters. "We are thousands here, where should we go?"
Hungary's ruling centre-right Fidesz party, led by Prime Minister Viktor Orban, has struck a combative tone in the migrant crisis. Antal Rogan, the Fidesz party's parliament caucus leader, said on Tuesday "the very existence of Christian Europe" was under threat
"Would we like our grandchildren to grow up in a United European Caliphate? My answer to that is no," Rogan told the pro-government daily Magyar Idok.
Rogan criticised EU leaders, including Commission President Jean-Claude Juncker, for what he said was an overly lenient attitude to migration.
"Will Mr. Juncker be there at the funeral of the illegal migrants who died in the truck (found in Austria last week)? Because it is irresponsible statements like his that allow human traffickers to get ill-fated people to do anything."
Orban's chief of staff, Janos Lazar, told a Parliament committee on Tuesday that immigration must be controlled tightly.
"I do not think Hungary would need a single immigrant from Africa or the Middle East," Lazar said. "Europe must use its own human resources fundamentally and if it wants an immigration policy it must be regulated and controlled."
"In the past decade... a leftist view has dominated the European Commission and the European Parliament, that the way to develop Europe was through allowing everyone in and accepting everyone without checks, rules and controls." (Reporting by Budapest bureau; Writing by Marton Dunai; Editing by Peter Graff)