Short Answer
Is government shutdown US specific?
In a very narrow sense, yes. Other countries don't usually shutdown due to the failure of the parliament to approve a budget (for reasons discussed here). But, in a somewhat broader sense, similar things do happen elsewhere.
There are somewhat analogous situations in other systems, such as the failure to form a government in Northern Ireland that resulted in layoffs at government contractors whose contracts could not be renewed as a result, but it doesn't play out in precisely the same way.
More often, the same practical effect that the U.S. sees from a government shutdown, due to its failure to approve appropriations bills, arises in other countries in connection with general strikes or public sector union strikes, which are very rare in the U.S. at the national level.
Long Answer
Failure To Form A Government
The closest analogy in a system that is parliamentary rather than Presidential, like the U.S., is when no party receives a majority of the seats in parliament and negotiations between the parties fail to produce a majority coalition that can form a government.
One of the most recent and severe examples of that kind of deadlock is from Belgium, called the 2007-2011 Belgian political crisis. Northern Ireland, similarly has had extended periods in which a government was not formed. Italy has also recently had this problem.
The practical effect of such a failure to form a government is particular to the government involved, and while a U.S. style federal government shutdown isn't common (and wasn't what happened in the U.S. prior to 1980 even when appropriations bills weren't passed in a timely manner), there are negative consequences and slow downs of key government functions when this happens. Usually, what happens is that the previous government continues to operate as an "interim government" until the new one is formed with the formal or informal constitutional understanding that the lame duck government does not have the authority to make any major policy changes after losing an election, and the status quo continues.
In Northern Ireland, for example, this meant that relatively routine public works contracts could not be authorized resulting in many construction workers who usually work on public works projects being laid off.
Another way that this situations are resolved is that the largest party in parliament forms a minority government which can be voted out anytime a majority can be organized against it, forcing it to tread carefully in building support and usually collapsing earlier than usual when some issue or another gives rise to a no confidence vote.
Some newer constitutions, such as the Constitution of Afghanistan adopted in 2004, often have default rules keeping some sort of government budget in place in circumstances when no majority can agree on a new budget.
General and Public Sector Strikes
This doesn't mean that other countries don't have periods of time when large numbers of public employees aren't working. But, more often, this is due to strikes by public sector workers or "general strikes" of all unionized employees in a country.
Public employee strikes happen when a government employer and its unions reach a deadlock in negotiations similar to the deadlock that has occurred involving Congress and the President in the U.S. in budget negotiations. General strikes usually concern government actions that have become wildly unpopular among a large cross-section of the population causing the labor movement generally to coordinate and organize a strike over broad political issues rather than over narrow employment contract issues.
Most U.S. government employees are forbidden from going on strike (although "sick outs" such as those currently going on in the TSA in the U.S. aren't unknown), and federal government employee strikes whether legal or not have been exceedingly rare in practice.
Wikipedia notes just six general strikes in U.S. history, one in 1919 in Seattle, and four in 1934, at the height of the Great Depression, one in Toledo, one in Minneapolis, one in San Francisco and one at West Coast Ports. There was also a general strike twenty years ago in the Commonwealth of Puerto Rico in 1998. The last one in the English Speaking United States was 76 years ago, and very few people living today remember it. The United States has never had a nationwide general strike and just two general strikes in its history, one restricted to a single industry, and the other to a U.S. territory, extended beyond a single city.
Per the same source, Spain has had four national general strikes since 2010 in addition to one European-wide general strike in 2012, and Italy had a national general strike in 2002. In 1995, Canada has a national general strike, and France had widespread public sector strikes. France is noted for its frequent public employee strikes that shut down government services in one sector or another. For example, this past summer, France finally concluded a four month long set of rolling strikes by rail workers.
