Please - what is this hen's diet? And does she have a constant supply of calcium available (e.g. oyster shell)?
The above posts are correct - if she doesn't get good help soon, she will likely die a painful death. As noted above, there are common household items that could help lubricate (mineral oil, vegetable oil). If none of those are in the house, please get them somehow - from a neighbor or whatever. It is troubling if the only folks who drive in your household are not willing to get a being in pain and distress what it needs; it is truly unfortunate that you are alone in this.
It has been recommended in other threads on BYC that in an emergency a Tums be given to help provide the calcium needed to get the egg out (assuming that's the problem here).
JJ
Edited to add that I am wondering if some aspirin in her water (FIVE 325 mg aspirins [NO SUBSTUTUES] crushed per gallon of water) might relieve enough pain to help her get the egg out, if indeed that is the problem. I would like it if someone else weighed in on this first.