Of course, the author meant administering eye drops. How do you change the phrase in order to exclude the undesirable meaning (burying people)?
Actually, the original phrase is not fully correct.
'To administer eye drops to patients' should be 'закапывать больным' with 'patients' in dative rather than accusative ('закапывать больных'). The accusative with 'закапывать' indicates whatever you use as drops (cf. 'закапывать капли'), so it would mean 'to drip patients' - or, indeed, 'to bury them', since the pres.perf.cont.(?) of 'копать'/dig and 'капать'/drip are the same.
Still, this phrasing is understandable, and close enough to existing hospital jargon for the joke to be funny.
So, one way to rephrase would be to make it grammatically correct, changing accusative to dative: 'Всем больным закапывать в семь утра' -but of course there are a ton of other ways to say this.