So in other words you are willing to engage in NO actual basic research on your own and merely are questioning my word for the heck of it. (You have provided zero evidence by the way that things in this area are different this time, especially since it makes sense conservative media would not focus on such a story.)
Here is some general info about contractors and the government.
http://govcon.mofo.com/acquisition-regulations/contractors-should-prepare-for-potential-shutdown/
Basically while it gets a bit complicated, the general key is what are legally considered non-essential services generally get shut down quickly, and temporary civilian contractors to deal with a shortage of chaplains of a particular denomination are particularly impacted. (While there is allot of talk in the link I provided about basically whether the contracting company can get damages from the sudden shutdown, this often does not impact if the individual employees can actually work during this period, or those employees only get paid if the contractor is nice and or they can get reassigned to non-government work, with the contractor paying them to be nice not meaning they can legally act as basically government employees during the shutdown.) If you look at your article, part of the issue was shut down government facilities and bases to the more general public, and legally speaking the priests became unauthorized personnel until the shutdown ended. Notably if you look at the details of the article, the priests were not actually barred from voluntarily ministering to the soldiers if the soldiers went off base, so the prohibition was by no means truly that complete if you really look at how it worked in practice.
Basically the issue is that essential personnel apply to those who are involved in national defense in a very direct way such as all active military members and civilian personnel where the military is going to have functional problems in the short term if they are furloughed and not available. The other exceptions apply to general health and safety or very basic urgent government functions of agencies on even a temporary basis. Contract priests do not fit in the same way, especially since the article you linked to basically indicated that spiritual counselling was in fact still being provided during the furlough especially if someone was feeling suicidal (although they generally are required to take specific steps in that scenario) or otherwise felt the need for counselling. The short version is any regular military chaplain (who by the way is an active member of the US military) is basically required to provide spiritual (and some degree of effective mental) counselling to any member of the military who walks through their office door, they can not be selective based on that soldier's personal religious beliefs and they are expected to provide counselling that can be useful even for a soldier that is a different religion.
Now especially in relation to bigger bases, the military does tend to try to provide chaplains to cover the major faiths of those who are serving at that base to a degree. (With protestant denominations this often ends up only partially covering the variations due to practical limitations of the burden of having too many chaplains at a base, but this means the chaplains are supposed to be broadly accommodating with their religious services for example.) What happened here was a select limited number of Catholic priests were temporary hired to deal with an issue where there were viewed as not enough to fully accommodate all the Catholic soldiers at various bases, but with a shutdown they could not really be justified as being truly essential so they had to be furloughed which legally limited what they could do. (To put things perspective, those in the military with less commonly practiced religions have to deal with the reality that there is no military chaplain who can explicitly really accommodate their particular religious practices most if not all the time.)