The first thing to note is that if the price increases are caused by supply shocks (as is almost certainly the case), higher beef prices are unambiguously bad news for consumers. So, even if consumers are today eating "healthier", they aren't happier. If people really wanted to be eating more veggies or chicken, they could have voluntarily chose to do so five years ago without a beef price spike forcing them into that outcome.
The core of the question relates to substitutability. When the price of beef rises, what do people substitute toward? This is the so-called cross-price elasticitiy of demand. There is an enormous academic literature in agricultural economics estimating the cross-price elasticity of demand for beef and other types of food.
The problem with a lot of this literature is that, due to data limitations, all beef is lumped together as an aggregate category and the researchers study how consumers substitute aggregate beef for aggregate pork and aggregate poultry. Most of the studies I've seen put the cross-price elasticities between aggregate beef and aggregate pork/poultry at something less than 0.3 and often much smaller than that. So, a 1% increase in the price of beef causes a less than 0.3% increase in pork and poultry. A 100% increase in the price of beef (something like what we've seen over the last 5 years) causes a less than 30% increase in quantity of pork an poultry demanded.
There are a few studies that look at how demand food categories is affected by aggregate meat (lumping together beef, pork, and poultry). This study, for example, estimates that a 1% increase in the price of meat would cause a 1% increase in quantity demanded for eggs, a 0.32% increase in demand for fruits and vegetables, a 0.24% increase in alcoholic beverages, and smaller effects on other categories of foods.
But, most of these types of studies only study the inter-related demands for different aggregate meats and do not include effects on fruits and vegetables or other foods. In so doing, the researchers are assuming beef demand is "separable" from demand for other foods. Separability is a bit of a complicated concept but it basically means that my demand for one good is not directly affected by the price of another good (though it may well be indirectly affected as I have to re-allocated my budget to different categories in light of the price increase).
There are some studies that have relaxed this separability assumption to look how consumers substitute among different types of meat. For example, when the price of steak rises, are consumers more likely to substitute toward another beef product (like roast or ground beef) or similar product from a different species like a pork chop or chicken breast? This older paper suggests in fact suggests that the latter is the case:
In any event, one of the big challenges with answering these sorts of questions is that we simply don't have good data. We can get aggregate "disappearance" data from the USDA which lumps all beef together. We can get grocery store scanner data, but most of that is of limited use for fresh meat products or fresh veggies because these are "random weight" items and the datasets don't have information on product weights (it's tough to estimate demand when you don't know exactly how much people are buying). That's one of the reasons I've been running the Food Demand Survey (FooDS) survey for over two years now, to provide better data to answer these questions. My colleague Glynn Tonsor (whose written a lot on the very issues I mention here) from K-State and I are polishing up a paper now that uses the FooDS data to look at substitution patters among different types of meat and non-meat items in this period of high beef prices, but it's not quite ready for preview yet.